First order ODE parameter fit to dataset

Hello,

I am trying to find the optimal parameters for my first order ODE to fit a dataset.

My ODE takes the following form: ds/dt= (alfa * temp - s(t)) * (1/tau)

The parameters that I need to find are "alfa" and "tau" whereas "temp" (temperature) depends on time but I have no equation to represent it, just a dataset of its evolution over time during the same timeframe of "ds/ dt" evolution.

I would appreciate any feedback on how to get "temp" into the code as my working version is not working. Please see below the code and its errors.

Many thanks in advance for any feedback.

CODE:

%Initial data

load rcp85_expansionmidmatlab.txt;
load rcp85_temperaturemidmatlab.txt;

time= rcp85_temperaturemidmatlab(:,1)-2006+1;
temp= rcp85_temperaturemidmatlab(:,2);
sealevel=rcp85_expansionmidmatlab(:,2);
plot(time,sealevel)

%ODE information
tSpan = [1 95];
z0 = [0.0118];
tempi=temp(tSpan);

%Initial guess
alfa=0.2;%lower range from Mengel paper
tau=82;%lower range from Mengel paper

ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau,tempi),tSpan,z0); % Run the ODE
simsealevel = deval(ODE_Sol, time); % Evaluate the solution at the experimental time steps

hold on
plot(time, simsealevel, '-r')

%% Set up optimization
myObjective = @(x) objFcn(x,time,sealevel,tSpan,z0);
lb = [0.2,82];
ub = [0.63,1290];

bestalfatau = lsqnonlin(myObjective, alfa,tau, lb, ub);


%% Plot best result
ODE_Sol = ode45(@(t,z)updateStates(t,z,bestalfatau,tempi),tSpan, z0);

bestsealevel = deval(ODE_Sol, time);

plot(time, bestsealevel, '-g')
legend('IPCC Data','Initial Param','Best Param');


function f = updateStates(t,z,alfa,tau,tempi)


f = (alfa.*tempi-z)*(1/tau);


function cost= objFcn (x,time,sealevel,tSpan,z0)


ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);
simsealevel = deval(ODE_Sol, time);


cost = simsealevel-sealevel;

ERRORS:

Error using odearguments (line 95)
@(T,Z)UPDATESTATES(T,Z,ALFA,TAU,TEMPI) returns a vector of length 2, but the length of initial conditions vector is 1.
The vector returned by @(T,Z)UPDATESTATES(T,Z,ALFA,TAU,TEMPI) and the initial conditions vector must have the same number
of elements.

Error in ode45 (line 115)
  odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in odeparam1 (line 21)
ODE_Sol= ode45(@(t,z)updateStates(t,z,alfa,tau,tempi),tSpan,z0); % Run the ODE
 

Answer 1

[Are Mjaavatten]

The challenge is how to use your table of temperatures in your updateStates function. At a given time t, you must interpolate your table to find the correct value. Here is a simple example:

T = [22,24,18,21];  % temperature series
ts =[0,5,10,20];    % Times when T is sampled
x0 = 16;
fun = @(t,x) interp1(ts,T,t)-x;
[t,x] = ode45(fun,[0,20],x0);
plot(t,x,ts,T)

Hopefully, you will be able to use this trick to solve your ODE. The next step is to of course to find your parameters. If you have access to the optimization toolbox, try fmununc or lsqnonlin.

[Maria Alvarez]

Thank you Are!

Following your suggestion I have modified "tempi" to this:

tempi=interp1(time,temp,tSpan);

I still get errors but will try to sort them through.

[Are Mjaavatten]

I suggest you define tempi as an inline function:

tempi=@(t) interp1(time,temp,t);

Next, modify updateStates to call tempi at time t:

function f = updateStates(t,z,alfa,tau,tempi)
    f = (alfa.*tempi(t)-z)*(1/tau);
end

[Maria Alvarez]

Thanks again, Are!

I made the suggested change and managed to get the simulated sea level - "simsealevel"- as curve on the plot.

However, I had to modify the "alfa" and "tau" parameters to a vector "c" to avoid some errors resulting from interactions between the lower and upper bounds of each.

I am not sure if as a result of these changes or what "tempi" is not recognised as a variable any longer.

Any pointers on where I am failing to represent this properly would be much appreciated.

Please see below for the code and errors.

CODE:

function bestalfatau = odeparam2()
%Initial data

load rcp85_expansionmidmatlab.txt;
load rcp85_temperaturemidmatlab.txt;

time= rcp85_temperaturemidmatlab(:,1)-2006+1;%modifying the vector from [2006 2100] to [1 95] to match tSpan
temp= rcp85_temperaturemidmatlab(:,2);
sealevel=rcp85_expansionmidmatlab(:,2);
plot(time,sealevel)

%ODE information
tSpan = [1 95];
z0 = [0.0118];
tempi=@(t) interp1(time,temp,t);

%Initial guess
c0= [0.2,82];


ODE_Sol= ode45(@(t,z)updateStates(t,z,c0,tempi),tSpan,z0); % Run the ODE
simsealevel = deval(ODE_Sol, time); % Evaluate the solution at the experimental time steps

hold on
plot(time, simsealevel, '-r')

%% Set up optimization
myObjective = @(x) objFcn(x,time,sealevel,tSpan,z0);
lb = [0.2,82];
ub = [0.63,1290];

bestc = lsqnonlin(myObjective,c0, lb, ub);


%% Plot best result
ODE_Sol = ode45(@(t,z)updateStates(t,z,bestc,tempi),tSpan, z0);

bestsealevel = deval(ODE_Sol, time);

plot(time, bestsealevel, '-g')
legend('IPCC Data','Initial Param','Best Param');


function f = updateStates(t,z,c,tempi)


f = (c(1).*tempi(t)-z)*(1/(c(2)));


function cost= objFcn (x,time,sealevel,tSpan,z0)


ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);
simsealevel = deval(ODE_Sol, time);


cost = simsealevel-sealevel;

ERRORS:

>> odeparam2
Unrecognized function or variable 'tempi'.

Error in odeparam2>objFcn (line 53)
ODE_Sol = ode45(@(t,z)updateStates(t,z,x),tempi, tSpan, z0);

Error in odeparam2>@(x)objFcn(x,time,sealevel,tSpan,z0) (line 28)
myObjective = @(x) objFcn(x,time,sealevel,tSpan,z0);

Error in lsqnonlin (line 206)
            initVals.F = feval(funfcn{3},xCurrent,varargin{:});

Error in odeparam2 (line 32)
bestc = lsqnonlin(myObjective,c0, lb, ub);

Caused by:
    Failure in initial objective function evaluation. LSQNONLIN cannot continue.

[Are Mjaavatten]

I made some modifications to make the code run with my dummy data. I have added some comments to most of my changes.

function bestc = odeparam2()
  %Initial data
  time = linspace(1,95,10);
  temp = [20,21,23,22,27,28,23,22,23,25];
%   sealevel=rcp85_expansionmidmatlab(:,2);
  sealevel = sin(time);
  plot(time,sealevel)
  %ODE information
  tSpan = [1 95];
  z0 = [0.0118];
  tempi=@(t) interp1(time,temp,t); 
  %Initial guess
  c0= [0.2,82];
  c0= [0.2,20];
  
  % I find it easier to interpret the code it I define fun outside
  % the call to ode45, bur this is a matter of taste
  fun = @(t,z) updateStates(t,z,c0,tempi); 
  ODE_Sol= ode45(fun,tSpan,z0); % Run the ODE
  simsealevel = deval(ODE_Sol, time); % Evaluate the solution at the experimental time steps
  hold on
  plot(time, simsealevel, '-r')
  %% Set up optimization
  myObjective = @(x) objFcn(x,time,sealevel,tempi,tSpan,z0);
%   lb = [0.2,82];
%   ub = [0.63,1290];
%   bestc = lsqnonlin(myObjective,c0, lb, ub);
%  The boundaries gave some errors that I did not bother to investigate
  bestc = lsqnonlin(myObjective,c0);
  %% Plot best result
  ODE_Sol = ode45(@(t,z)updateStates(t,z,bestc,tempi),tSpan, z0);
  bestsealevel = deval(ODE_Sol, time);
  plot(time, bestsealevel, '-g')
  legend('IPCC Data','Initial Param','Best Param');
end

function f = updateStates(t,z,c,tempi)
  f = (c(1).*tempi(t)-z)*(1/(c(2)));
end
function cost= objFcn (x,time,sealevel,tempi,tSpan,z0)
%   disp(x)  % to follow the iteration process
  fun = @(t,z) updateStates(t,z,x,tempi);
  ODE_Sol = ode45(fun, tSpan, z0);
  simsealevel = deval(ODE_Sol, time);
  ds = simsealevel-sealevel;
  cost = ds*ds';    % The objective must be a scalar
end

With my data, the objective was almost insenitive to tau for small values of alfa, so lsqnonln did not vary tau. With your data, it may be different. Here is how I visualised myObjective:

alfa = 0.001:0.001:0.02;
tau = 10:5:80;
[A,T] = meshgrid(alfa,tau);
r = zeros(size(A));
for i = 1:15;for j = 1:20;r(i,j) = myObjective([A(i,j),T(i,j)]);end;end
figure;surf(alfa,tau,r)
xlabel \alpha;ylabel \tau;zlabel objective

[Maria Alvarez]

Thanks very much, Are!